The obstruction theories defined above in \(D^b(\mathcal {M}(\tilde{X},\beta ^{\prime })_x)\)  and \(D^b(\mathcal {M}(X,\beta )_x)\)  are perfect obstruction theories for the two horizontal morphisms. Since we have seen that the obstruction theory on \(\mathcal {M}(X,\beta )_x\)  pulls back to the obstruction theory on \(\mathcal {M}(\tilde{X},\beta ^{\prime })_x\) , and the morphism \(\mathfrak {M}^{^{\prime }ev}(\tilde{\mathcal {X}} \rightarrow \mathcal {X})_x \rightarrow \mathfrak {M}^{ev}(\mathcal {X})_x\)  is degree 1, the same argument as in proof of Theorem \(1.1.1\)  of [1]}An error in this argument was amended by Herr and Wise in [2]} gives \(\pi _*([\mathcal {M}(\tilde{X},\beta ^{\prime })_x]^{vir}) = [\mathcal {M}(X,\beta )_x]^{vir}\) .
